Other than passes allowing unlimited travel, do any train tickets allow one to get off before reaching their destination then continuing on later?
In Germany, you can specify a stopover for several hours on a Super Saver Ticket (advance purchase nonrefundable) on national trains (ICE, IC, EC). I forget the time limit, but I believe it can be until the last train of the day.
If you continue on a regional train, you can take any regional train after the national train. Must be shown on the original ticket. The journey is completed by 10 am the next day.
If you have a super saver fare ticket, you can only travel on the service indicated on the ticket. The offers are available for ICE, IC/EC connections within Germany. You can use local rail services (e.g. RE, RB, IRE and S-Bahn or non-DB trains) to get to/from the long-distance station if you include them in your booking. You are not restricted to travelling on a specific train when using local rail services. However, your journey must be completed by 10 am the following day.
If you take only regional trains, you can stop off anywhere along the route, and continue the journey on any regional train.
If you are traveling in a given Land (roughly equivalent to a US state), you can buy a Laender ticket and travel on any regional train in that Land until 3 am the next day. You have to travel weekday after 9 am, or anytime weekends and holidays. In a big Land like Bavaria, that can be very useful. That also includes local transport like U-bahns, trams, buses.
In Switzerland, you can stop over anywhere along the route as long as you do not have a discounted train-specific ticket.
You don't specify which country you are speaking of.
In the UK there are some tickets which allow in one direction, some in both, and many not at all. It is important to be sure you have the correct ticket - there are many.
Regional trains in Italy have some flexibility, generally good for 4 hours from the time you validate the ticket in the platform stamping machine. Since the trains are subsidized by the provincial governments, these rules can vary depending on where you are in Italy. So short stopovers are possible.
TER trains in France are good for 24 hours from the time you validate your ticket in the platform stamping machine. At least that was the case 10 years ago, but I can't find the current conditions of carriage on line.
Belgian trains are generally good for a stopovers, except for high speed trains like the Thalys. Also, there is a surcharge for stopping at Brussels airport.
Netherlands trains run like a suburban network where you need to scan your ticket to enter and exit the station platform, so stop overs are problematic, but then the cost of buying individual tickets is not a problem.
.....passes allowing unlimited travel...... Not sure what you mean by that or your understanding. Generally speaking -- in broad terms because every train system is different -- a train pass does not convey the right jump on any train at any time. Often the train will require a seat reservation -- at an additional cost -- along with the train pass. So if you got off the train you would need a new seat reservation to get back on later. However, some trains system run a mix of reserved and non-reserved seats so you could do that on those trains. You might have to stand also if you cannot find an open seat. So without knowing which countries you are traveling through, the response can only be --- It depends.
And do the math. A rail pass may or may not be the most economic choice. These days the advance discount prices for tickets may under cut a pass.
Just out of curiosity, I queried the Bahn schedule and fare webpage (reiseankunft.bahn.de/bin/query.exe/en) for a connection leaving Munich after 14:00 going to Frankfurt Flughafen with a 17 hour stopover in Würzburg.
The first connection the Bahn offered was leaving Munich at 14:45, arriving FRA at 14:45 the next day. It only offered one fare, a flexipreis (standard) ticket for 98,50 EUR. When I selected "earlier", it offered me a connection leaving Munich at 12:45, arriving at Würzburg at 14:52. The stopover in Würzburg was 17 hr, 32 min, leaving Würzburg at 8:24 the next morning, arriving at FRA at 9:56. For that connection, it offered both a flexipreis of 98,50 EUR and a Super Sparpreis fare of 23,90 EUR (also a Sparpreis fare of 29,90).
If you take only regional trains, you can stop off anywhere along the
route, and continue the journey on any regional train.
I think the same 100km rule applies to regional tickets (less than 100km, finish by end of day; more than 100km, by end of 2nd day). On a regional pass, you must finish your travel by 3 AM the next morning.
I've tried the same thing multiple times in the past, putting in a stopover giving me time for an overnight stay, with similar results. My advice, don't ask us here, try it yourself with an inquiry on the Bahn website.
BTW, if you tried that connection with a railpass, it would use two days of your pass.